Imagine a spherical shell having charge Q. The Electric Field exists outside. I bring a test charge from infinity to the surface of the shell (I bring it radially to simplify the calculation and moreover the field is conservative) and solve the Integral to get KQ/a (a is the radius). Now I move the charge inside but there is no field there. Mathematically I divide the Integral in 2 parts , 1st from infinity to a and 2nd from a to b. The Integral from a to b gives 0 and hence the result.
Now imagine two concentric spherical shells of radius a and b. The outer shell carries charge -2Q (radius a) and the inner carries Q. By the above method of bringing the test charge from infinity to the point, I get the potential at the surface of outer sphere as -kQ/a (since field outside is -kQ/(r^2). For the potential at the surface of the inner sphere (by the method above ) , I divide the Integral from infinity to a and from a to b. Now the 1st integral is mathematically just the just the above case (case of potential at the surface of outer sphere) and gives -kQ/a and since the field inside (between the 2 spheres) is only due to Q , I get kQ/b. Thus total potential at b is -kQ/a+ kQ/b. But this is wrong, we know it has to be -2kQ/a + kQ/b.
So where I am going wrong ?